The hypersurface in the sphere (Pursuit of the Essence of Singularity Theory)

The hypersurface

in

the

sphere

Yang

Jiang

School

_of

Mathematics

and

Statistics, Northeast Normal University

Abstract

We consider hypersurfaces in the unit lightlike sphere. The unit sphere can be

canonically embedded in the lightcone and de Sitter space in Minkowski space. We

investigate thesehypersurfaces intheframeworkofthe theory of Legendriandualities

betweenpseudo-spheresinMinkowski space. This isan anouncement of the results in [15]

1.

Introduction

In [2, 3], professor Izumiya has introduced the mandala of Legendrian dualities between pseudo-spheres in Minkowski space. There are three kinds of pseud$(\succ$spheres in Minkowski

space (i.e.,Hyperbolicspace,deSitter space and thelightcone). Especially, if

we

investigate

spacelike submanifolds in the lightcone, those Legendrian dualities are essentially useful (see, also [7]). Forde Sitter spaceand thelightcone in Minkowski $(n+2)$

-space,

there exist

naturally embedded unit $r\succ$-spheres. Moreover,

we

have the canonical projection from the

lightcone to the unit sphereembedded in the lightcone(cf.,

_{\S 2).}

In thispaper weinvestigate hypersurfaces in the unit $n$-sphere in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski $(n+2)$-space ([3, 4, 12, 13], etc.). If we have a

hypersurface in the unit $\gamma\triangleright$sphere, then we have spacelike hypersurfaces in the embedded

unit $n$-sphere in the lightcone and de Sitter space. Therefore, we naturally have the dual hypersurfaces in the lightcone as an application of the duality theorem in [3]. There are

two kinds of lightcone dual hypersurfaces ofahypersurface in the unit $n$-sphere. One is the dual ofthe hypersurface ofthe unit $n$-sphere embedded in de Sitter space and another is

the dualofthe hypersurface of theunit $n$-sphereembedded in the lightcone. By definition, these dual hypersurfaces are different.

On the other hand,

we

have

studied

the

curves

in the unit 2-sphere and the unit 3-sphere

from the view point of the Legendrian duality in [5, 6]. In the unit 2-sphere, it is known

that the evolute of

a curve

in the unit 2-sphere is the dual of the tangent indicatrix of the

original curve [11]. We have shown that the projection images of the critical value sets of lightcone dual surfaces for a curve in the unit 2-sphere coincide with the evolute of the original curve in [5]. However, this fact doesn’t hold for a curve in unit 3-sphere (cf., [6]). For the curve case, these facts has been shown by the direct calculations in [5, 6]. We have not known the geometric

reason

why the situations are different. In order to clarify these situation, weinvestigatehypersurfaces in the unit $n$-sphere from the viewpoint ofthe

theory ofLegendrian singularities. Thecurves in theunit 2-sphere can be considered as a

special

case

ofthis paper. We

can

also show that the projection images of the critical value sets of two different lightcone dual hypersurfaces for a hypersurface in the unit $n$-sphere also coincide with the spherical evolute (cf., [10]) of the original hypersurface. We interpret

geometric meanings of the singularities of those two lightcone dual hypersurfaces. Here, we remark that we do not have the notion of tangent indicatrices for higher dimensional submanifolds in the sphere. Therefore, the situation is completely different from the curve

case. In [15], we givea classification of thegeneric singularities ofthe lightcone duals of the

surface in the unit 3-sphere.

All maps and submanifolds considered here areofclass $C^{\infty}$ unless otherwise stated.

2.

The

basic

concepts

Let$\mathbb{R}^{n+2}$bean_$(n+2)$-dimensional vector space. Forany

two vectors$x=(x_{0}, x_{1}, \ldots, x_{n+1}),$ _$y=$ $(y_{0}, y_{1}, \ldots, y_{n+1})$ in $\mathbb{R}^{n+2}$, their pseudo scalar

product is defined by $\langle x,$$y\rangle=-x_{0}y_{0}+$

$x_{1}y_{1}+\ldots+x_{n+1}y_{n+1}$. Here, $(\mathbb{R}^{n+2}, \langle)\rangle)$ is called Lorentz-Minkowski $(n+2)$-space (simply,

Minkowski$(n+2)$-space),which is denoted by$\mathbb{R}_{1}^{n+2}$.Forany $(n+1)$vectors

$x_{1},$ $x_{2},$_$\ldots,$$x_{n+1}\in$

$\mathbb{R}_{1}^{n+2}$, their pseudo vector product is defined by

$x_{1}\wedge x_{2}\wedge\ldots\wedge x_{n+1}=$

$-e_{0}$ $e_{1}$ . . . _{$e_{n+1}$} $x_{1}^{0}$ $x_{1}^{1}$ . .

.

$x_{1}^{n+1}$

$x_{2}^{0}$ $x_{2}^{1}$

.

.

.

$x_{2}^{n+1}$

. . .

$x_{n+1}^{0}$ $x_{n+1}^{1}$ . .

.

$x_{n+1}^{n+1}$

where $\{e_{0}, e_{1}, \cdots, e_{n+1}\}$ is the canonical basis of $\mathbb{R}_{1}^{n+2}$ and _{$x_{i}=(x_{i}^{0},x_{i}^{1}, \cdots,x_{i}^{n+1})$}. $A$

non-zerovector $x\in \mathbb{R}_{1}^{n+2}$ is calledspacelike, lightlike or timelike if $\langle x,$$x\rangle>0,$$\langle x,$$x\rangle=0$ or

$\langle x,$$x\rangle<0$ respectively. The norm of$x\in \mathbb{R}_{1}^{n+2}$ is defined by $\Vert x\Vert=\sqrt{|\langle x,x\rangle|}$

.

We define the de Sitter $(n+1)$-space by

$S_{1}^{n+1}=\{x\in \mathbb{R}_{1}^{n+2}|\langle x, x\rangle=1\}.$

We define the dosed lightcone with the vertex$a$ by

$LC_{a}=\{x\in \mathbb{R}_{1}^{n+2}|\langle x-a,x-a\rangle=0\}.$

We define the open lightcone at the onginby

$LC^{*}=\{x\in \mathbb{R}_{1}^{n+2}\backslash \{0\}|\langle x, x\rangle=0\}.$

We consider a submanifold in the lightcone defined by $S_{+}^{n}=\{x\in LC^{*}|x_{1}=1\}$, which is

called the lightlike unit sphere. We have a projection $\pi$ : _{$LC^{*}arrow S_{+}^{n}$} defined by

$\pi(x)=\tilde{x}=(1,\frac{x_{1}}{x_{0}}, \ldots, \frac{x_{n+1}}{x_{0}})$ ,

where$x=(x_{0}, x_{1}, \ldots x_{n+1})$.We also define the$n$-dimensional Euclideanunitsphere in$\mathbb{R}_{0}^{n+1}$

by $S_{0}^{n}=\{x\in S_{1}^{n+1}|x_{0}=0\}$, where $\mathbb{R}_{0}^{n+1}=\{x\in \mathbb{R}_{1}^{n+2}|x_{0}=0\}.$

Let $x$ : $Uarrow S_{+}^{n}$ be an embedding from an open set $U\subset \mathbb{R}^{n-1}$

.

We identify $M=x(U)$

with $U$ through the embedding _$x$

.

Obviously, the tangent space

$T_{p}M$ are all spacelike (i.e.,

consists only spacelike vectors), so $M$ is a spacelike hypersuface in $S_{+}^{n}\subset \mathbb{R}_{1}^{n+2}$

.

We have

a map $\Phi$ :

$S_{+}^{n}arrow S_{0}^{n}$ defined by $\Phi(v)=v-e_{0}$, which is an isometry. Then we have a

hypersurface$\overline{x}$: _{$Uarrow S_{0}^{n}$} defined by

same

geometric properties

as

spherical hypersurfaces. For any $p=x(u)$, we

can

construct a unit normal vector$n(u)$

as

$n(u)= \frac{\overline{x}(u)\wedge e_{0}\wedge x_{u_{1}}(u)\wedge..\cdot\wedge x_{u_{n-1}}(u)}{\Vert\overline{x}(u)\wedge e_{0}\wedge x_{u_{1}}(u)\wedge..\wedge x_{u_{n-1}}(u)\Vert}.$

We have $\langle n(u),$$n(u)\rangle=1,$ $\langle e_{0},$$e_{0}\rangle=-1$ and $\langle e_{0},$$n\rangle=\langle n,x_{u_{i}}\rangle=\langle n,x\rangle=0$

.

The system

$\{e_{0},n(u),\overline{x}(u),x_{u_{1}}(u), \ldots, x_{u_{n-1}}(u)\}$is a basisof $l_{p}’\mathbb{R}_{1}^{n+2}$

.

We define

a

map_{$G:Uarrow S_{0}^{n}$}

by$G(u)=n(u)$. We call it the Gauss mapofthe hypersurface$M=x(U)$. We have alinear mapping provided by the derivation of the Gauss map at $p\in M,$ $dG(u)$ : $T_{p}Marrow T_{p}M.$

We call the linear transformation$S_{p}=-dG(u)$ the shape operatorof $M$ at $p=x(u)$

.

The

eigenvalues of$S_{p}$ denoted by $\{\kappa_{i}(p)\}_{i=1}^{n-1}$ are called the principal curvatures of$M$ at $p$

.

The

Gauss-Kronecker curvatureof$M$ at$p$is defined to be$K(p)=\det S_{p}.$ $A$ point$p$is called

an

umbdic point ifall theprincipal curvatures coincide at $p$ andthus wehave $S_{p}=\kappa(p)id_{1_{p}M}$

forsome $\kappa(p)\in \mathbb{R}$

.

We saythat$M$is totallyumbilic ifallthe points on$M$areumbilic. Since

$x$ is aspacelike embedding, we have a$Ri$emannian metric (or the

first

fundamenta)form)

on

$M$ given by $ds^{2}= \sum_{i,j=1}^{n-1}g_{ij}du_{i}du_{j}$, where $g_{ij}(u)=\langle x_{u_{i}}(u),$$x_{u_{j}}(u)\rangle$ for any $u\in U.$

The secondfundamental form on $M$ is given by$h_{ij}(u)=-\langle n_{u_{t}}(u),$$x_{u_{j}}(u)\rangle$ at any $u\in U.$

Under the above notation, we have the following Weingarten formula [9]:

$G_{u_{t}}=- \sum_{j=1}^{n-1}h_{i}^{j}x_{u_{j}}(i=1, \ldots, n-1)$,

where $(h_{i}^{j})=(h_{ik})(g^{kj})$ and $(g^{kj})=(g_{kj})^{-1}$. This formula induces an explicit expression

of the Gauss-Kroneckercurvature in terms of the Riemannianmetricand the second

funda-mentalinvariant given by $K=det(h_{ij}/det(g_{\alpha\beta})$

.

$A$ point _$p$is

a

parabolic point if$K(p)=0.$

Apoint $p$is a

flat

pointif it is

an

umbilicpoint and $K(p)=0.$

In [10] the spherical evolute of a hypersurface has been introduced and investigated the singularities. Each spherical evolute of$\overline{M}=\overline{x}(U)$ isdefined to be

$\epsilon\frac{\pm}{M}=\bigcup_{i=1}^{n-1}\{\pm(\sqrt{\frac{\kappa_{t}^{2}(p)}{1+\kappa_{i}^{2}(p)}}\overline{x}(u)+\sqrt{\frac{1}{1+\kappa_{i}^{2}(p)}}n(u))|p=x(u)\in M=x(U)\}.$

3.

The lightcone dual surfaces and the

lightcone height

functions

In [3], professor Izumiya hasintroducedtheLegendrian dualitiesbetween pseudo-spheres

in Minkowski space which is a basic tool for the study of hypersurfaces in pseudo-spheres

in Minkowski space. We define one-forms $\langle dv,$$w \rangle=-w_{0}dv_{0}+\sum_{i=1}^{n+1}w_{i}dv_{i},$ $\langle v,$$dw\rangle=$

$-v_{0}dw_{0}+ \sum_{i=1}^{n+1}v_{i}dw_{i}$

on

$\mathbb{R}_{1}^{n+2}\cross \mathbb{R}_{1}^{n+2}$ and consider the following two double fibrations:

(1)(a) $LC^{*}\cross S_{1}^{n+1}\supset\Delta_{3}=\{(v, w)|\langle v, w\rangle=1\},$ (b) $\pi_{31}:\triangle_{3}arrow LC^{*},\pi_{32}:\Delta_{3}arrow S_{1}^{n+1},$

(c) $\theta_{31}=\langle dv,$$w\rangle|\Delta_{3},$$\theta_{32}=\langle v,$$dw\rangle|\Delta_{3}.$

(2) (a) $LC^{*}\cross LC^{*}\supset\Delta_{4}=\{(v, w)|\langle v, w\rangle=-2\},$ (b) $\pi_{41}:\Delta_{4}arrow LC^{*},\pi_{42}:\Delta_{4}arrow LC^{*},$ (c) $\theta_{41}=\langle dv,$$w\rangle|\Delta_{4},$$\theta_{42}=\langle v,$$dw\rangle|\Delta_{4}.$

Here, $\pi_{i1}(v, w)=v,$ $\pi_{i2}(v, w)=w$

.

We remark that $\theta_{i1}^{-1}(0)$ and $\theta_{i2}^{-1}(0)$ define the same

that each $(\Delta_{i}, K_{i})(i=3,4)$ is a contact manifold and both of$\pi_{ij}(j=1,2)$ areLegendrian

fibrations. Moreover thosecontact manifolds are contact diffeomorphic to each other. In [3]

we have defined four double fibrations $(\triangle_{i}, K_{i})(i=1,2,3,4)$ such that these are contact

diffeomorphic to each other. Here, we only use $(\triangle_{3}, K_{3})$ and $(\triangle_{4}, K_{4})$

.

If we have an

isotropic mapping $i$ : $Larrow\Delta_{i}$ $(i.e., i^{*}\theta_{i1}=0)$, we say that $\pi_{i1}(i(L))$ and $\pi_{i2}(i(L))$ are

$\Delta_{i}$-dual to each other $(i=3,4)$

.

For detailed properties ofLegendrian fibrations, see [1].

Wenow define hypersurfaces in $LC^{*}$ associated with the hypersurfaces in $S_{+}^{n}$ or $S_{0}^{n}$

.

Let

$x$ : $Uarrow S_{+}^{n}$ be ahypersurface. We define$\overline{LD}\frac{\pm}{M}$ : _{$U\cross \mathbb{R}arrow LC^{*}$} by

$\overline{LD}\frac{\pm}{M}(u,\mu)=\overline{x}(u)+\mu n(u)\pm\sqrt{\mu^{2}+1}e_{0}.$

We also define $LD_{M}$ : $U\cross \mathbb{R}arrow LC^{*}$ by

$LD_{M}(u, \mu)=(\mu^{2}/4-1)\overline{x}(u)+\mu n(u)+(\mu^{2}/4+1)e_{0}.$

Then we have the following proposition.

Proposition 3.1. Under the above notation, we have the followings: (1) $\overline{x}$ and$\overline{LD}\frac{\pm}{M}$ are $\Delta_{3}$-dualto each other.

(2) $x$ and$LD_{M}$ are$\Delta_{4}$-dual to each other.

We call each one of$\overline{LD}\frac{\pm}{M}$ the lightcone dual hypersurface along

$\overline{M}\subset S_{0}^{n}$ and $LD_{M}$ the

lightconedual hypersurface along$M\subset S_{+}^{n}$

.

Thenwehavetwo mappings

$\pi 0\overline{LD}\frac{\pm}{M}$

: $U\cross \mathbb{R}arrow$

$S_{+}^{n}$ and $\pi oLD_{M}$ : $U\cross \mathbb{R}arrow S_{+}^{n}$ defined by

$\pi\circ\overline{LD}\frac{\pm}{M}(u, \mu) = \pm(\frac{1}{\sqrt{\mu^{2}+1}}\overline{x}(u)+\frac{\mu}{\sqrt{\mu^{2}+1}}n(u))+e_{0},$

$\pi\circ LD_{M}(u, \mu) = \frac{\mu^{2}-4}{\mu^{2}+4}\overline{x}(u)+\frac{4\mu}{\mu^{2}+4}n(u)+e_{0}.$

Let $x$ : $Uarrow S_{+}^{n}$ be a hypersurface in the lightlike unit sphere. Then we define two

families offunctions as follows:

$\overline{H}$ : _{$U\cross LC^{*}arrow \mathbb{R}$};

$\overline{H}(u,\overline{v})=\langle\overline{x}(u),\overline{v}\rangle-1,$

$H:U\cross LC^{*}arrow \mathbb{R}$; $H(u, v)=\langle x(u),$$v\rangle+2.$

We$cal1\overline{H}$alightcone height

function

of the deSitterspherical hypersurface M. For anyfixed

$\overline{v}_{0}\in LC^{*}$, wedenote $\overline{h}_{\overline{v}_{0}}(u)=\overline{H}(u, \overline{v}_{0})$. We also call $H$ a $light\omega ne$ height

function

ofthe lightlike spherical hypersurface $M$

.

For any fixed $v_{0}\in LC^{*}$, wedenote $h_{v_{0}}(u)=H(u, v_{0})$

.

Proposition 3.2. Let $\overline{M}$ be a hypersurface in

$S_{0}^{n}$ and$\overline{H}$ the lightcone height

funct\’ion

on

M.

For$p=x(u)$ and$\overline{p}=\overline{x}(u)\neq\overline{v}^{\pm}$, we have the followmgs:

(1) $\overline{h}_{\overline{v}}\pm(u)=\partial\overline{f}_{b_{v}}\pm/\partial u_{i}(u)=0(i=1, \ldots,n-1)$\’ifand only

if

$\overline{v}^{\pm}=\overline{LD}\frac{\pm}{M}(u, \mu)$

for

some $\mu\in \mathbb{R}\backslash \{0\}.$

(2} $\overline{h}_{\overline{v}}\pm(u)=\partial\overline{h}_{\overline{v}}\pm/\partial u_{i}(u)=0(i=1, \ldots, n-1)$ and$\det$ Hess $(\overline{h}_{\overline{v}}\pm)(u)=0$

if

and only

if

$\overline{v}^{\pm}=\overline{LD}\frac{\pm}{M}(u, \mu),$ _$1/\mu$ is one

Proposition 3.3. Let$M$ be a hypersurface in$S_{+}^{n}$ and $H$bethe lightcone height

function

on

M. For$p=x(u)\neq v$, we _{have the follounngs.}

(1) $h_{v}(u)=\partial h_{v}/\partial u_{i}(u)=0,$ $(i=1, \ldots,n-1)$

if

andonly

if

$v=LD_{M}(u, \mu)$

for

some $\mu\in \mathbb{R}\backslash \{0\}.$

(2) $h_{v}(u)=\partial h_{v}/\partial u_{i}(u)=0,$ _{$(i=1, \ldots,n-1)$} and $\det$ Hess _{$(h_{v})(u)=0$}

if

and only

if

$v=LD_{M}(u, \mu),$ $(\mu/4-1/\mu)w$ one _the non-zero _{principle curvatures} $\kappa_{i}(p)$

of

$M.$

Let $(u, \mu)$ be

a

singularpoint of each

one

of$\overline{LD}\frac{\pm}{M}$

.

By Proposition 3.2,

we

have $1/\mu=$

$\kappa_{i}(p)(1\leq i\leq n-1)$, where $\kappa_{i}(p)$ is one of the

non-zero

principle curvatures of _$M$ at

$p=x(u)$. It follows that $\mu=1/\kappa_{i}(p)$

.

Therefore the critical value sets of$\overline{LD}\frac{\pm}{M}$ are given

by

$C( \overline{LD}\frac{\pm}{M})=\bigcup_{i=1}^{n-1}\{\overline{x}(u)+\frac{1}{\kappa_{i}(p)}n(u)\pm\sqrt{\frac{1}{\kappa_{i}^{2}(p)}+1}e_{0}|u\in U\}.$

Let $(u, \mu)$ be a singular point of$LD_{M}(u, \mu)$

.

By Proposition 3.3, we have $\mu/4-1/\mu=$

$\kappa_{i}(p)(1\leq i\leq n-1)$. It follows that wehave$\mu=2(\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)})$

.

For simplification,

we write that $\sigma^{\pm}(\kappa_{i}(p))=\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)}$. Then the critical value sets of _{$LD_{M}$} are given by

$C(LD_{M})^{\pm}= \bigcup_{i=1}^{n-1}\{((\sigma^{\pm}(\kappa_{i}(p)))^{2}-1)\overline{x}(u)+2\sigma^{\pm}(\kappa_{i}(p))n(u)+((\sigma^{\pm}(\kappa_{i}(p)))^{2}+1)e_{0}|u\in U\}.$

We respectively denote that

$LF \frac{\pm}{M}=\bigcup_{i=1}^{n-1}\{\overline{x}(u)+\frac{1}{\kappa_{i}(p)}n(u)\pm\sqrt{\frac{1}{\kappa_{i}^{2}(p)}+1}e_{0}|u\in U\},$

$LF_{M}^{\pm}= \bigcup_{i=1}^{n-1}\{((\sigma^{\pm}(\kappa_{i}(p)))^{2}-1)\overline{x}(u)+2\sigma^{\pm}(\kappa_{i}(p))n(u)+((\sigma^{\pm}(\kappa_{i}(p)))^{2}+1)e_{0}|u\in U\}.$

We respectively call each one of $LF^{\underline{\pm}}$ the lightcone

focal surface

ofthe de Sitter spherical

hypersurface $\overline{x}$ and each one of $LF_{M}\ovalbox{\tt\small REJECT}$

the ligtcone

_{focal surface}

of the lightcone spherical hypersurface$x$. Then the projections of these surfaces to $S_{+}^{n}$

are

given

as

follows:

$\pi(C(\overline{LD}\frac{\pm}{M}))=\bigcup_{i=1}^{n-1}\{\pm(\sqrt{\frac{\kappa_{i}^{2}(p)}{1+\kappa_{i}^{2}(p)}}\overline{x}(u)+\sqrt{\frac{1}{1+\kappa_{i}^{2}(p)}}n(u))+e_{0}|u\in U\},$

$\pi(C(LD_{M})^{\pm})=\bigcup_{i=1}^{n-1}\{\frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}-1}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1}\overline{x}(u)+\frac{2\sigma^{\pm}(\kappa_{i}(p))}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1}n(u)+e_{0}|u\in U\}.$

Bydefinition, wehave$\epsilon\frac{\pm}{M}=\Phi\circ\pi(C(\overline{LD}\frac{\pm}{M}))$, whereeach oneof

$\epsilon\frac{\pm}{M}$is thespherical evolute

sets of thelightconedual hypersurfacesof$\overline{M}=\overline{x}(U)$.Since$\sigma^{\pm}(\kappa_{i}(p))=\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)},$ wehave$(\sigma^{\pm}(\kappa_{i}(p)))^{2}=2\kappa_{i}(p)\sigma^{\pm}(\kappa_{i}(p))+1$

.

By straightforward calculations, we have

$( \frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}-1}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1})^{2}=\frac{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}}{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}+(\sigma^{\pm}(\kappa_{i}(p)))^{2}}=\frac{\kappa_{i}^{2}(p)}{1+\kappa_{i}^{2}(p)}$

and

$( \frac{2\sigma^{\pm}(\kappa_{i}(p))}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1})^{2}=\frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}}{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}+(\sigma^{\pm}(\kappa_{i}(p)))^{2}}=\frac{1}{1+\kappa_{i}^{2}(p)}.$

Thus we have the following proposition.

Proposition 3.4. Let$x$ : $Uarrow S_{+}^{n}$ be ahypersurface in$S_{+}^{n}$

.

Then

We define$\tilde{\pi}=\Phi 0\pi$ : _{$LC^{*}arrow S_{0}^{n}$}

.

Then we havethe following theorem

as

acorollary of Proposition 3.4.

Theorem 3.5. Both

_of

the projections

_of

the critical volue sets $C( \overline{LD}\frac{\pm}{M})$ and _{$C(LD_{M})^{\pm}$}

in the unit sphere $S_{0}^{n}$ are the images

of

the sphericalevolutes

of

$M.$

$\tilde{\pi}(C(\overline{LD}\frac{\pm}{M}))=\tilde{\pi}(C(LD_{M})^{\pm})=\epsilon\frac{\pm}{M}.$

4.

The lightcone dual hypersurfaces

as

wave

fronts

Wenow naturally interpret the lightcone dual hypersurfaces of the submanifolds in $S_{+}^{n}$ as

wavefrontsets in the theory of Legendrian singularities. Let $\overline{\pi}$:$PT^{*}(LC^{*})arrow LC^{*}$ be the

projective cotangentbundleswith canonicalcontactstructures. Considerthetangent bundle

$\tau$ : $TPT^{*}(LC^{*})arrow PT^{*}(LC^{*})$ and the differential map $d\overline{\pi}:TPT^{*}(LC^{*})arrow T(LC^{*})$ of

$\overline{\pi}$

.

For any

$X\in TPT^{*}(LC^{*})$, there exists an element $\alpha\in T^{*}(LC^{*})$ such that $\tau(X)=[\alpha].$

For an element $V\in 1_{v}(LC^{*})$, the property $\alpha(V)=0$ dose not depend on the choice of

representative of the class $[\alpha]$

.

Thus we have the canonical contact structure on_{$PT^{*}(LC^{*})$}

by

$K=\{X\in TPT^{*}(LC^{*})|\tau(X)(c\Gamma\pi(X))\}=0.$

coordinate neighborhood $(U, (\pm\sqrt{v_{1}^{2}++v_{n+1}^{2}}, v_{1}, \ldots,v_{n+1}))$ in $LC^{*}$, we have a

trivial-ization $PT^{*}(LC^{*})\equiv LC^{*}\cross P(\mathbb{R}^{n})^{*}$ and we call $((\pm\sqrt{v_{1}^{2}++v_{n+1}^{2}}, v_{1}, \ldots, v_{n+1}),$ $[\xi_{1}$ :

. .

.

_: $\xi_{n+1}])$ homogeneous coordinates of$PT^{*}(LC^{*})$, where $[\xi_{1} :. . .: \xi_{n+1}]$ arethe homoge-neous coordinates of the dual projective space $P(\mathbb{R}^{n})^{*}$

.

It is easy to show that _{$X\in K_{(v,[\xi])}$}

if and only if $\sum_{1}^{n+1}\mu_{i}\xi_{i}=0$, where $c f\overline{\pi}(X)=\sum_{1}^{n+1}\mu_{i}\partial/\partial v_{i}\in 1_{v}’LC^{*}$. An immersion $i:Larrow PT^{*}(LC^{*})$ issaid to be aLegendrianimmersionif$\dim L=n$and$\dot{d}i_{q}(T_{q}L)\subset K_{i(q)}$ for any $q\in L$

.

The map To$i$ is also called the Legendrian map and we call the set $W(i)=image\overline{\pi}\circ i$ thewave front of$i$. Moreover, $i$(

lift of $W(i)$. Let $F$ : $(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}, 0)$ be a function germ. We say that $F$ is a Morse family of hypersurfaces if the map germ $\Delta^{*}F$ : $(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}^{k+1},0)$ defined by $\Delta^{*}F’=(F, \partial F/\partial u_{1}, \cdots, \partial F/\partial u_{k})$

.

is nonsingular. In this case, we have the following

smooth $(n-1)$-dimensional smoothsubmanifold.

$\Sigma_{*}(F)=\{(u, v)\in(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)|F(u, v)=\frac{\partial F}{\partial u_{1}}(u, v)=\cdots=\frac{\partial F}{\partial\eta 1_{k}}(u, v)=0\}=(\Delta^{*}F)^{-1}(0)$

.

The map germ $\mathcal{L}_{F’}$ : _{$(\Sigma_{*}(F’), 0)arrow P’1’*\mathbb{R}^{n}$} defined by

$\mathcal{L}_{F’}(u, v)=(v, [\frac{\partial F}{\partial v_{1}}(u, v)$ :.. . : $\frac{\partial F}{\partial v_{n}}(u, v)])$

.

is a Legendrian immersion germ. Then we have the following fundamental theorem of

Amol’d and Zakalyukin [1, 14].

Proposition4.1. AllLegendrian

_submanifold

germsin$PT^{*}\mathbb{R}^{n}$ areconstructed by the above

method.

We call $F$ a generatingfamily of$\mathcal{L}_{F}(\Sigma_{*}(F))$

.

Therefore the wave front of$\mathcal{L}_{F}$ is

$W(\mathcal{L}_{F})=\{v\in \mathbb{R}^{n}|\exists u\in \mathbb{R}^{k}$such that $F(u, v)= \frac{\partial F}{\partial u_{1}}(u, v)=\ldots=\frac{\partial F}{\partial u_{k}}(u, v)=0\}.$ We claim here that we have atrivialization

as

follows:

$\Phi$ : _{$PT^{*}(LC^{*}) \equiv LC^{*}\cross P(\mathbb{R}^{n})^{*};\Phi([\sum_{i=1}^{n+1}\xi_{i}dv_{i}])=(v_{0}, v_{1}, \cdots, v_{n+1}),$} $[\xi_{1} :. .. \xi_{n+1}])$ by using the above coordinate system.

Proposition 4.2. The lightcone height

_function

$H$ : $U\cross LC^{*}arrow \mathbb{R}$ is a Morse famdy

of

the hypersurface around $(u, v)\in\Sigma_{*}(H)$

.

We also have the following proposition.

Proposition 4.3. The lightcone height

_function

$\overline{H}$ : _{$U\cross LC^{*}arrow \mathbb{R}$} is a

Morse $fam\iota ly$

of

the hypersurface around $(u, v)\in\Sigma_{*}(\overline{H})$. Here,

we

consider the Legendrian immersion

$\mathcal{L}_{4}$ : $(u, \mu)arrow\triangle_{4};\mathcal{L}_{4}(u, \mu)=(LD_{M}(u, \mu),x(u))$

.

We define the following mapping:

$\Psi$ :

$\Delta_{4}arrow LC^{*}\cross P(\mathbb{R}^{n})^{*};\Psi(v, w)=(v, [v_{0}w_{1}-v_{1}w_{0} :. . . :v_{0}w_{n+1}-v_{n+1}w_{0}])$

.

For the canonical contact form $\theta=\sum_{i=1}^{n+1}\xi_{i}dv_{i}$ on $PT^{*}(LC^{*})$, we have $\Psi^{*}\theta=(v_{0}w_{1}-$ $v_{1}w_{0})dv_{1}+\cdots+(v_{0}w_{n+1}-v_{n+1}w_{0})dv_{n+1}|_{\Delta_{4}}=v_{0}(-w_{0}dv_{0}+w_{1}dv_{1}+\cdots+w_{n+1}dv_{n+1})-$ $w_{0}(-v_{0}dv_{0}+v_{1}dv_{1}+\cdots+v_{n+1}dv_{n+1})|_{\Delta_{4}}=v_{0}\langle w,$$dv\rangle|_{\Delta_{4}}=v_{0}\theta_{42}|_{\triangle_{4}}$. Thus $\Psi$ is a contact

morphism.

Theorem 4.4. For any hypersurface $x$ : $Uarrow S_{+}^{n}$, the lightcone height

function

$H$ : $U\cross LC^{*}arrow \mathbb{R}$ is agenerating family

of

the Legendnan immersion $\mathcal{L}_{4}.$

Similarly,weconsider theLegendrianimmersions$\mathcal{L}_{3}^{\pm}:(u,\mu)arrow\Delta_{3}$ definedby$\mathcal{L}_{3}^{\pm}(u,\mu)=$ $( \overline{LD}\frac{\pm}{M}(u, \mu),\overline{x}(u))$

.

Then wehave the following theorem.

Theorem 4.5. For any hypersurface $\overline{x}$ : _{$Uarrow S_{0}^{n}$}, the lightcone height

function

$\overline{H}$ : $U\cross LC^{*}arrow \mathbb{R}$ is agenerating family

of

the Legendrian immersions $\mathcal{L}_{3}^{\pm}.$

5.

Contact

with parabolic

$(n-1)$

-spheres

and parabolic

$n$

-hyperquadrics

Beforewe startto consider thecontact between hypersurfaces in the sphere with parabolic

$(n-1)$-sphere and parabolic $n$-hyperquadrics, we briefly reviewthe theory ofcontact due

to Montaldi[8]. Let $X_{i},$$Y_{i}(i=1,2)$ be submanifolds of $\mathbb{R}^{n}$ with

$\dim X_{1}=\dim X_{2}$ and

$\dim Y_{1}=\dim Y_{2}$

.

We say that the contact of$X_{1}$ and$Y_{1}$ at $y_{1}$ is thesame typeas the $\omega$ntact

of

$X_{2}$ and$Y_{2}$ at$y_{2}$ if thereisa diffeomorphism$\Phi$ : $(\mathbb{R}^{n}, y_{1})arrow(\mathbb{R}^{n}, y_{2})$ such that $\Phi(X_{1})=X_{2}$ and $\Phi(Y_{1})=Y_{2}$. In this case, we write $K(X_{1}, Y_{1};y_{1})=K(X_{2}, Y_{2};y_{2})$

.

Of course, in the

definition, $\mathbb{R}^{n}$ canbereplaced by any manifold. Two function germs

$f_{i}$ : $(\mathbb{R}^{n}, a_{i})arrow \mathbb{R}(i=$

$1,2)$ arecalled $\mathcal{K}$-equivalent if there is a diffeomorphismgerm $\Phi$ : _{$(\mathbb{R}^{n}, a_{1})arrow(\mathbb{R}^{n}, a_{2})$}, and

a function germ $\lambda$ : _{$(\mathbb{R}^{n}, a_{1})arrow \mathbb{R}$} with _{$\lambda(a_{1})\neq 0$} such that

$f_{1}=\lambda\cdot(f_{2}o\Phi)$

.

Theorem5.1 (Montaldi[8]). Let$X_{i},$ $Y_{i}$$(for i=1,2)$ be

submanifol& of

$\mathbb{R}^{n}$ with$dimX_{1}=dimX_{2}$

and $dimY_{1}=dimY_{2}$. Let $g_{i}$ : $(X_{i}, x_{i})arrow(\mathbb{R}^{n}, y_{i})$ be immersiongerms and$f_{i}$ : $(\mathbb{R}^{n}, y_{i})arrow$

$(\mathbb{R}^{p}, 0)$ besubmersiongermswith$(Y_{i}, y_{i})=(f_{i}^{-1}(0), y_{i})$. Then$K(X_{1}, Y_{1};y_{1})=K(X_{2}, Y_{2};y_{2})$

if

and $07dy$

if

$f_{1}og_{1}$ and$f_{2}og_{2}$ are $\mathcal{K}$-equivalent.

Retuming to the lightcone dual hypersurface $LD_{M}$, we

now

consider the function $\mathfrak{h}$ :

$S_{+}^{n}\cross LC^{*}arrow \mathbb{R}$ defined by $\mathfrak{h}(u, v)=\langle u,$$v\rangle+2$ and the function $\mathfrak{g}$ : $LC^{*}\cross LC^{*}arrow \mathbb{R}$

defined by $\mathfrak{g}(u, v)=\langle u,$ $v\rangle+2$ . For a given $v_{0}\in LC^{*}$, we denote $\mathfrak{h}_{v_{0}}(u)=\mathfrak{h}(u, v_{O})$ and $\mathfrak{g}_{v_{U}}(u)=\mathfrak{g}(u, v_{0})$, then we have$\mathfrak{h}_{v_{0}}^{-1}(0)=S_{+}^{n}\cap HP(v_{O}, -2)$ and$\mathfrak{g}_{v_{0}}-1(0)=LC^{*}\cap HP(v_{O}, -2)$

.

Forany $u_{0}\in U,$ $\mu_{0}\in \mathbb{R}$, wetake thepoint _{$v_{0}=LD_{M}(u_{0}, \mu_{0})$}. Then we have

$\mathfrak{g}_{v_{0}}ox(u_{0})=\mathfrak{g}o(x\cross id_{LC^{*}})(u_{0}, v_{0})=\mathfrak{h}_{v_{0}}ox(u_{0})=\mathfrak{h}\circ(x\cross id_{LC^{*}})(u_{0}, v_{0})=H(u_{0}, v_{0})=0.$

We also have

$\frac{\partial(\mathfrak{g}_{v_{0}}\circ x)}{\partial u_{i}}(u_{0})=\frac{\partial(\mathfrak{h}_{v_{0}}\circ x)}{\partial u_{i}}(u_{0})=\frac{\partial H}{\partial u_{i}}(u_{0}, v_{0})=0$

for$i=1,$ $\cdots,$$n-1$

.

This meansthat the $(n-1)$-sphere$\mathfrak{h}_{v_{0}}^{-1}(0)=S_{+}^{n}\cap HP(v_{0}, -2)$ istangent

to $M=x(U)$ at$p_{0}=x(u_{0})$. Inthis case, we call it the lightcone tangent parabolic _$(n-1)-$

sphere of $M$ at $p_{0}$, which is denoted by $TPS_{+}^{n-1}(x, u_{0})$. The $n$-hyperquadric $\mathfrak{g}_{v_{0}}^{-1}(0)=$

$LC^{*}\cap HP(v_{O}, -2)$ is alsotangent to $M$ at $p_{0}$

.

In thiscase, wecall it the lightcone tangent

parabolic$n$-hyperquadric of $M$ at $p_{0}$, which is denoted by $TPH^{}$ $(x, u_{0})$. Forthe lightcone

dual surfaces $\overline{LD}\frac{\pm}{M}$, we consider a function $\overline{\mathfrak{h}}$ :

$S_{0}^{n}\cross LC^{*}arrow \mathbb{R}$ defined by $\overline{\mathfrak{h}}(u, v)=$ $\langle u,$$v\rangle-1$ and a function $\overline{\mathfrak{g}}.$ $S_{1}^{n+1}\cross LC^{*}arrow \mathbb{R}$ defined by $\overline{\mathfrak{g}}(u, v)=\langle u,$$v\rangle-1$ For a

given $v_{0}\in LC^{*}$, we denote that $\overline{\mathfrak{h}}_{v_{0}}(u)=\overline{\mathfrak{h}}(u, v_{O})$ and $\overline{\mathfrak{g}}_{v0}(u)=\overline{\mathfrak{g}}(u, v_{0})$

.

Then we have

$\overline{\mathfrak{h}_{v0}}^{1}(0)=S_{0}^{n}\cap HP(v_{O}, 1)$ and $\overline{\mathfrak{g}_{v_{0}}}^{1}(0)=S_{1}^{n+1}\cap HP(v_{0},1)$

.

For any _{$u_{0}\in U$} and the points

$\overline{v}_{0}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0}, \mu_{0})$, wehave

$\overline{\mathfrak{g}}_{\overline{v}_{\cup}}\pm\circ\overline{x}(u_{0})=\overline{\mathfrak{g}}\circ(\overline{x}\cross id_{LC^{*}})(u_{0}, \overline{v}_{0}^{\pm})=\overline{\mathfrak{h}}_{\overline{v}_{\cup}}\pm\circ\overline{x}(u_{0})=\overline{\mathfrak{h}}\circ(\overline{x}\cross id_{LC^{*}})(u_{0}, \overline{v}_{0}^{\pm})=\overline{H}(u_{0}, \overline{v}_{0}^{\pm})=0.$

We also have

$\frac{\partial(\overline{\mathfrak{g}}_{\overline{v}_{0}^{\pm}}\circ\overline{x})}{\partial u_{i}}(u_{0})=\frac{\partial(\overline{\mathfrak{h}}_{\overline{v}_{0}^{\pm}}\circ\overline{x})}{\partial v_{i}}(u_{0})=\frac{\partial\overline{H}}{\partial u_{i}}(u_{0}, \overline{v}_{0}^{\pm})=0$

for$i=1,$$\cdots,$$n-1$. It follows that eachoneof the $(n-1)-$sphere$\overline{\mathfrak{h}_{\overline{v}_{0}}}^{1}\pm(0)=S_{0}^{n}\cap HP(\overline{v}_{0}^{\pm}, 1)$

is tangent to $\overline{M}$ at

$\overline{p}_{0}=\overline{x}(u_{0})$

.

In this case, we call each one the tangent parabolic

$(n-1)$-sphere of$\overline{M}$

the n-hyperquadric$\overline{\mathfrak{g}}_{\overline{v}_{0}^{\pm}}^{-1}(0)=S_{1}^{n+1}\cap HP(\overline{v}_{0}^{\pm}, 1)$ is tangent to $\overline{M}$at

$\overline{p}_{0}$

.

In this case,

we

call

each onethe de-Sitter tangent parabolic $n$-hyperquadric of$\overline{M}$at

$\overline{p}_{0}$, which are denoted by

$TPS_{1}^{n\pm}(\overline{x},u_{0})$.

Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs. For $v_{l}\prime=LD_{M_{t}},$$(u_{i}, \mu_{i})$,

we denote $h_{i,v_{i}}$ : $(U, u_{i})arrow(\mathbb{R}, 0)$ by $h_{i,v_{i}}(u_{i})=H(u_{i}, v_{i})$

.

Then

we

have $h_{i,v_{t}}(u)=$

$(\mathfrak{h}_{i,v_{i}}ox_{i})(u)=(\mathfrak{g}_{i,v_{i}}ox_{i})(u)$

.

For$\overline{v}_{i}^{\pm}=\overline{LD}\frac{\pm}{M}i(u_{i}, \mu_{i})$, We denote

$\overline{h}_{i,\overline{v}_{t}}\pm:(U, u_{i})arrow(\mathbb{R}, 0)$

by $\overline{h}_{i,\overline{v}_{i}^{\pm}}(u_{i})=\overline{H}(u_{i},\overline{v}_{i}^{\pm})$

.

Then wehave $\overline{h}_{i,\overline{v}_{i}}\pm(u)=(\overline{\mathfrak{h}}_{i,\overline{v}_{i}}\pm\circ\overline{x}_{i})(u)=(\overline{\mathfrak{g}}_{i,\overline{v}_{i}}\pm\circ\overline{x}_{i})(u)$

.

By

Theorem 5.1, we have the following proposition.

Proposition 5.2. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs. For $v_{i}=$

$LD_{M_{:}}(u_{i},\mu_{i})$, the follovring conditions are equivalent:

(1) $K(x_{1}(U),TPS_{+}^{n-1}(x_{1}, u_{1}), v_{1})=K(x_{2}(U),TPS_{+}^{n-1}(x_{2}, u_{2}), v_{2})$

.

(2) $K(x_{1}(U),TPH^{n}(x_{1}, u_{1}), v_{1})=K(x_{2}(U),$TPH $(x_{2}, u_{2}),$$v_{2})$.

(3) $h_{1,v_{1}}$ and$h_{2,v_{2}}$

are

$\mathcal{K}$-equivalent.

Moreover,

_for

$\overline{v}_{it\prime}^{\pm_{=\overline{LD}\frac{\pm}{M}}}(u_{i},\mu_{i})$, thefollounng

$\omega$nditions

are

equivalent:

(4) $K(x_{1}(U)_{)}TPS_{0}^{n-1\pm}(x_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(x_{2}(U), TPS_{0}^{n-1\pm}(x_{2}, u_{2})_{)}\overline{v}_{2}^{\pm})$.

(5) $K(x_{1}(U),TPS_{1}^{n\pm}(x_{1}, u_{1}),\overline{v}_{1}^{\pm})=K(x_{2}(U)_{)}TPS_{1}^{n\pm}(x_{2}, u_{2}), \overline{v}_{2}^{\pm})$ .

(6) $\overline{h}_{1,\overline{v}_{1}}\pm$ and$\overline{h}_{2,\overline{v}_{2}}\pm are\mathcal{K}$-equival$ent.$

On the otherhand, w\‘ereturnto the review

on

thetheory of Legendrian singularities. We

introduce a natural equivalence relation among Legendrian submanifold germs. Let $F,$ $G$ :

$(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}, 0)$ be Morse families of hypersurfaces. Then we say that $\mathcal{L}_{F}(\Sigma_{*}(F))$

and $\mathcal{L}_{G}(\Sigma_{*}(G))$ are Legendrian equivalent if there exists a contact diffeomorphism germ

$H$ : $(PT^{*}\mathbb{R}^{n}, z)arrow(PT^{*}\mathbb{R}^{n}, z’)$ such that$H$preserves fibersof$\pi$and that$H(\mathcal{L}_{F’}(\Sigma_{*}(F’)))=$

$\mathcal{L}_{G}(\Sigma_{*}(G))$, where $z=\mathcal{L}_{F’}(0),$$z’=\mathcal{L}_{G}(0)$

.

By using the Legendrian equivalence, we

can

definethe notion ofLegendrian stability for Legendrian submanifold germsby the ordinary way (see, [l][Part III]). We

can

interpret the Legendrian equivalence by usingthe notion of generating families. We denote by $\mathcal{E}_{n}$ the local ring offunction germs $(\mathbb{R}^{n}, 0)arrow \mathbb{R}$ with

the unique maximal ideal $\mathfrak{M}_{n}=\{h\in \mathcal{E}_{n}|h(O)=0\}$

.

Let $F,$$G:(\mathbb{R}^{k}\cross \mathbb{R}^{n},0)arrow(\mathbb{R}, 0)$be function germs.

Let $Q_{n+1}(x, u_{0})$ be thelocal ring of the function germ $h_{v_{0}}$ : $(U, u_{0})arrow \mathbb{R}$ defined by

$Q_{n+1}(x,u_{0})=C_{u_{U}}^{\infty}(U)/(\langle h_{v_{0}}\rangle_{C_{u}^{\infty}(U)}+\mathfrak{M}_{n-1}^{n+2})0,$

and $Q_{n+1}^{\pm}(\overline{x}, u_{0})$be the local rings of the function germs$\overline{h}_{\overline{v}_{0}^{\pm}}$ : $(U,u_{0})arrow \mathbb{R}$ defined by $Q_{n+1}^{\pm}(\overline{x}, u_{0})=C_{u_{0}}^{\infty}(U)/(\langle\overline{h}_{\overline{v}_{U}^{\pm}}\rangle_{C_{u_{0}}^{\infty}(U)}+\mathfrak{M}_{n-1}^{n+2})$,

where$v_{0}=LD_{M}(u_{0}, \mu_{0}),$$\overline{v}_{0}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0}, \mu_{0})$and

$C_{u0}^{\infty}(U)$is the localringoffunction germs

at $u_{0}$ with theuniquemaximal ideal $\mathfrak{M}_{n-1}.$

Theorem 5.3. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs such that

the $\omega$rresponding Legendrian immersion germs are Legendrian stable. Then the following

conditions are equivalent.

(1) The lightcone hypersurface germs $LD_{M_{1}}(U\cross \mathbb{R})$ and$LD_{M_{2}}(U\cross \mathbb{R})$ arediffeomorphic. (2) Legendrian immersiongerms$\mathcal{L}_{4}^{1}$ and $\mathcal{L}_{4}^{2}$ are Legendrian equivalent.

(3) The lightcone height

_functions

germs $H_{1}$ and$H_{2}$ are$\mathcal{P}-\mathcal{K}$-equivalent.

(4) $h_{1,v_{1}}$ and$h_{2,v}2$ are $\mathcal{K}$-equivalent.

(6) $K(x_{1}(U),$TPH $(x_{1}, u_{1}),$$v_{1})=K(x_{2}(U),$TPH $(x_{2}, u_{2}),$$v_{2})$.

(7) Local rings$Q_{n+1}(x_{1}, u_{1})$ and$Q_{n+1}(x_{2}, u_{2})$ are isomorphic as$\mathbb{R}$-algebras.

Theorem 5.4. Let $\overline{x}_{i}$ : $(U, u_{i})arrow(S_{0}^{n},p_{i})(i=1,2)$ be hypersurface germs such that

the $\omega$rresponding Legendrian immersion germs are Legendnan stable. Then the following

conditions are equivalent.

(1) The lightcone hypersurface germs$\overline{LD}\frac{\pm}{M}1(U\cross \mathbb{R})and\overline{LD}\frac{\pm}{M}2(U\cross \mathbb{R})$ are diffeomorphic.

(2) Legendrian immersion germs $\mathcal{L}_{3}^{1\pm}$ and $\mathcal{L}_{3}^{2\pm}$ are Legendrian equivalent.

(3) The lightcone height

_functions

germs $\overline{H}_{1}$ and$\overline{H}_{2}$ are$\mathcal{P}-\mathcal{K}$-equivalent.

(4) $\overline{h}_{1,\overline{v}_{1}}\pm$ and$\overline{h}_{2,\overline{v}_{2}}\pm$ are

$\mathcal{K}$-equivalent.

(5) $K(\overline{x}_{1}(U), TPS_{0}^{n-1\pm}(\overline{x}_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(\overline{x}_{2}(U), TPS_{0}^{n-1\pm}(\overline{x}_{2}, u_{2}), \overline{v}_{2}^{\pm})$

.

(6) $K(\overline{x}_{1}(U), TPS_{1}^{n\pm}(\overline{x}_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(\overline{x}_{2}(U), TPS_{1}^{n\pm}(\overline{x}_{2}, u_{2}),\overline{v}_{2}^{\pm})$ .

(7) Localrings $Q_{n+1}^{\pm}(\overline{x}_{1}, u_{1})$ and$Q_{n+1}^{\pm}(\overline{x}_{2},u_{2})$ are isomorphic as$\mathbb{R}$-algebras.

Lemma 5.5. Let $x$ : $Uarrow S_{+}^{n}$ be a hypersurface germ such that the $\omega$rresponding

Leg-endrian immersion germs $\mathcal{L}_{4}$ and $\mathcal{L}_{3}^{\pm}$

are

Legendrian stable. Then at the

singular point

$v_{0}=LD_{M}(u_{0},2\sigma^{\pm}(\kappa_{i}(p_{0})))(1\leq i\leq n-1)$

of

$LD_{M}$ and the $sing_{lJ}lar$ points $\overline{v}_{0}^{\pm}=$

$\overline{LD}\frac{\pm}{M}(u_{0},1/\kappa_{i}(p_{0}))of\overline{LD}\frac{\pm}{M}$, we have the following equivalent assertions:

(1) The lightcone hypersurface germs $LD_{M}(Ux\mathbb{R})$ and$\overline{LD}\frac{\pm}{M}(U\cross \mathbb{R})$ are diffeomorphic.

(2) Legendrian immersion germs $\mathcal{L}_{3}^{\pm}$ and$\mathcal{L}_{4}$ are Legendrian equivalent.

(3) Thelightcone height

_functions

germs $H$ and

i7

are $\mathcal{P}-\mathcal{K}$-equivalent.

(4) $h_{v_{0}}$ and$\overline{h}_{\overline{v}_{\cup}}\pm$ are

$\mathcal{K}$-equivolent.

(5) $K(x(U), TPS_{+}^{n-1}(x, u_{0}), v_{0})=K(\overline{x}(U), TPS^{n-1\pm}(\overline{x}, u_{0}),\overline{v}_{0}^{\pm})$

.

(6) $K(x(U),\prime 1’PH^{n}(x, u_{0}), v_{0})=K(\overline{x}(U), TPS_{1}^{n}(\overline{x}, u_{0}),\overline{v}_{0}^{\pm})\ovalbox{\tt\small REJECT}.$

(7) Local rings $Q_{n+1}^{\pm}(\overline{x}, u_{0})$ and_{$Q_{n+1}(x, u_{0})$} are oeomorphic as $\mathbb{R}$-algebras.

By Lemma 5.5, we have our mainresult as the following theorem.

Theorem 5.6. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs such that

the corresponding Legendrian immersion germs are Legendrian stable. At the singular points $\overline{v}_{i}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0},1/\kappa_{j}(p))(1\leqj\leq n-1)$

of

$\overline{LD}\frac{\pm}{M}$,

and the $sing\tau rlar$ points _{$v_{i}=$} $LD_{M}(u_{0},2\sigma^{\pm}(\kappa_{j}(p)))$

of

_{$LD_{M}$}, the $\omega$nditions ($1)\sim(7)$ in Theorem 5,3 and the conditions

(1) $\sim(7)$ in Theorem

5.4

are all equivalent.

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YANG JIANG

A. SCHOOL OF MATHEMATICS AND STATISTICS

NORTHEAST NORMAL UNIVERSITY

CHANGCHUN 130024 P. R. CHINA B. DEPARTMENT OF MATHEMATICS HOKKAIDO UNIVERSITY SAPPORO 060-OSIO JAPAN